Formula Library

Explore the formulas behind our calculators. Each entry includes the formula expression, variable definitions, and a worked example.

Includes: BMI ยท Compound interest ยท Mortgage payment ยท Percentage ยท Quadratic equation ยท Ohm's Law ยท Standard deviation ยท Newton's laws ยท and 190+ more.

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Monthly Mortgage Payment

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A mortgage calculator estimates monthly payments on a home loan by factoring in the principal amount, interest rate, loan term, property taxes, and insurance. It helps homebuyers understand total borrowing costs before committing to a mortgage.

M = P ร— [r(1+r)^n] / [(1+r)^n - 1]
SymbolVariableUnit
MMonthly PaymentUSD
PPrincipal Loan AmountUSD
rMonthly Interest Rateannual rate / 12
nTotal Number of Paymentsmonths

Worked Example

Home Price: $400,000Down Payment: 20% ($80,000)Interest Rate: 6.5%Loan Term: 30 years

Result: $2,023/month

Principal + interest only. Add property tax (~$350/mo) and insurance (~$125/mo) for total PITI of ~$2,498/mo.

Body Mass Index (BMI)

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A BMI (Body Mass Index) calculator measures body fat based on height and weight for adult men and women. BMI is a screening tool that categorizes individuals as underweight (<18.5), normal (18.5-24.9), overweight (25-29.9), or obese (30+).

BMI = weight (kg) / height (m)ยฒ
SymbolVariableUnit
BMIBody Mass Indexkg/mยฒ
weightBody weightkilograms
heightHeightmeters

Worked Example

Weight: 70 kg (154 lbs)Height: 1.75 m (5'9")

Result: BMI = 22.9 (Normal weight)

70 / (1.75 ร— 1.75) = 70 / 3.0625 = 22.86, which falls in the Normal weight category (18.5-24.9).

A percentage calculator finds the percentage of a number, calculates percentage increase/decrease, or determines what percentage one number is of another. It answers questions like "What is 15% of 200?" (Answer: 30) or "What percentage is 30 of 200?" (Answer: 15%).

Percentage = (Part / Whole) ร— 100
SymbolVariableUnit
PartThe portion or subsetany
WholeThe total or base amountsame as Part

Worked Example

Part: 45Whole: 200

Result: 22.5%

(45 / 200) ร— 100 = 22.5%. So 45 is 22.5% of 200.

Compound Interest

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A compound interest calculator computes how money grows when interest is earned on both the initial principal and previously accumulated interest. Unlike simple interest, compound interest accelerates wealth growth over time, making it the foundation of savings, investing, and debt calculations.

A = P(1 + r/n)^(nt)
SymbolVariableUnit
AFinal AmountUSD
PInitial PrincipalUSD
rAnnual Interest Ratedecimal
nCompounding Frequencytimes per year
tTimeyears

Worked Example

Principal: $10,000Rate: 7%Compounding: MonthlyTime: 20 years

Result: $40,387.39

$10,000 compounded monthly at 7% for 20 years grows to $40,387.39 โ€” that's $30,387.39 in interest earned.

Monthly Loan Payment

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A loan calculator estimates monthly payments, total interest, and amortization schedule for personal loans, auto loans, and other installment debt. It helps borrowers compare loan offers by showing the true cost of borrowing at different rates and terms.

PMT = P ร— [r(1+r)^n] / [(1+r)^n - 1]
SymbolVariableUnit
PMTMonthly PaymentUSD
PLoan PrincipalUSD
rMonthly Interest Rateannual rate / 12
nTotal Number of Paymentsmonths

Worked Example

Loan Amount: $25,000Interest Rate: 5.9%Loan Term: 5 years (60 months)

Result: $482.80/month

Total paid: $28,968. Total interest: $3,968. Effective cost of the loan is 15.9% of principal.

Common Area Formulas

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An area calculator computes the surface area of geometric shapes including rectangles, circles, triangles, trapezoids, and irregular polygons. It's used in construction, landscaping, real estate, and academic settings.

Rectangle: A = l ร— w | Circle: A = ฯ€rยฒ | Triangle: A = ยฝbh
SymbolVariableUnit
AAreasquare units
lLengthโ€”
wWidthโ€”
rRadiusโ€”
bBaseโ€”
hHeightโ€”

A concrete calculator estimates the volume of concrete needed for slabs, footings, walls, and columns. It converts dimensions into cubic yards or cubic meters and calculates the number of pre-mixed bags required, helping reduce waste and cost on construction projects.

Volume = Length ร— Width ร— Depth (+ 10% waste factor)
SymbolVariableUnit
VolumeConcrete neededcubic yards
LengthSlab lengthfeet
WidthSlab widthfeet
DepthSlab thicknessinches (convert to feet)

Worked Example

Length: 20 feetWidth: 10 feetDepth: 4 inches (0.333 ft)

Result: 2.47 cubic yards (+ 10% waste = 2.72 yards)

20 ร— 10 ร— 0.333 = 66.6 cu ft รท 27 = 2.47 cu yd. Order 10% extra for waste and spillage.

TDEE (Mifflin-St Jeor + Activity Factor)

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Total Daily Energy Expenditure โ€” the total number of calories your body burns in a day including basal metabolism, physical activity, and the thermic effect of food.

TDEE = BMR ร— Activity Factor
SymbolVariableUnit
BMR (men)10 ร— weight(kg) + 6.25 ร— height(cm) โˆ’ 5 ร— age โˆ’ 5kcal/day
BMR (women)10 ร— weight(kg) + 6.25 ร— height(cm) โˆ’ 5 ร— age โˆ’ 161kcal/day
Activity FactorSedentary 1.2 | Light 1.375 | Moderate 1.55 | Active 1.725 | Very Active 1.9multiplier

Worked Example

Weight: 80 kgHeight: 175 cmAge: 30Sex: MaleActivity: Moderate (1.55)

Result: 2,697 calories/day

BMR = 10(80) + 6.25(175) โˆ’ 5(30) โˆ’ 5 = 1,740. TDEE = 1,740 ร— 1.55 = 2,697 kcal.

Mifflin-St Jeor BMR

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Basal Metabolic Rate โ€” the number of calories your body needs at complete rest to maintain vital functions like breathing, circulation, and cell production.

BMR = 10W + 6.25H โˆ’ 5A + S
SymbolVariableUnit
WWeightkg
HHeightcm
AAgeyears
SSex constant+5 (men) or โˆ’161 (women)

Worked Example

Weight: 70 kgHeight: 165 cmAge: 25Sex: Female

Result: 1,396 calories/day

BMR = 10(70) + 6.25(165) โˆ’ 5(25) โˆ’ 161 = 700 + 1,031.25 โˆ’ 125 โˆ’ 161 = 1,445 (rounded).

US Navy Body Fat Method

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The proportion of your total body weight that is fat tissue, expressed as a percentage.

BF% (men) = 86.010 ร— logโ‚โ‚€(abdomen โˆ’ neck) โˆ’ 70.041 ร— logโ‚โ‚€(height) + 36.76
SymbolVariableUnit
abdomenWaist circumference at navelcm
neckNeck circumferencecm
heightHeightcm

Worked Example

Waist: 85 cmNeck: 38 cmHeight: 178 cm

Result: ~17.8% body fat

BF% = 86.010 ร— logโ‚โ‚€(85โˆ’38) โˆ’ 70.041 ร— logโ‚โ‚€(178) + 36.76 โ‰ˆ 17.8%.

Blood Alcohol Content โ€” the percentage of alcohol in a person's bloodstream, used to measure intoxication level.

BAC = (A / (W ร— r)) โˆ’ (0.015 ร— H)
SymbolVariableUnit
ATotal alcohol consumedgrams
WBody weightgrams
rWidmark factor0.68 (men) / 0.55 (women)
HHours since first drinkhours

Worked Example

Drinks: 3 beers (42 g alcohol)Weight: 80 kgSex: MaleTime: 2 hours

Result: 0.047% BAC

BAC = (42 / (80,000 ร— 0.68)) โˆ’ (0.015 ร— 2) = 0.077 โˆ’ 0.03 = 0.047%.

Pace Calculation

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The time it takes to cover one unit of distance while running or walking, typically expressed in minutes per mile or minutes per kilometer.

Pace = Total Time / Distance
SymbolVariableUnit
Total TimeDuration of run/walkminutes
DistanceDistance coveredmiles or km

Worked Example

Time: 28 minutesDistance: 3.1 miles (5K)

Result: 9:02 min/mile (5:36 min/km)

28 รท 3.1 = 9.03 min/mile. Convert to km: 28 รท 5 = 5.6 min/km.

The maximum weight an individual can lift for a single repetition of a given exercise with proper form.

1RM = Weight ร— (1 + Reps / 30)
SymbolVariableUnit
WeightWeight liftedlbs or kg
RepsRepetitions completedcount (โ‰ค10 for accuracy)

Worked Example

Weight: 185 lbsReps: 6

Result: 222 lbs estimated 1RM

1RM = 185 ร— (1 + 6/30) = 185 ร— 1.2 = 222 lbs.

Mosteller Formula

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Body Surface Area โ€” the measured or calculated surface area of the human body, used in medicine for drug dosing and burn assessment.

BSA (mยฒ) = โˆš(Height(cm) ร— Weight(kg) / 3600)
SymbolVariableUnit
HeightPatient heightcm
WeightPatient weightkg

Worked Example

Height: 170 cmWeight: 70 kg

Result: 1.82 mยฒ

BSA = โˆš(170 ร— 70 / 3600) = โˆš(3.306) = 1.818 mยฒ.

The state of consuming fewer calories than your body expends, leading to weight loss over time.

Deficit = TDEE โˆ’ Calorie Intake
SymbolVariableUnit
TDEETotal Daily Energy Expenditurekcal/day
Calorie IntakeDaily food intakekcal/day

Worked Example

TDEE: 2,500 kcalIntake: 2,000 kcal

Result: 500 kcal/day deficit โ†’ ~1 lb/week weight loss

1 lb of fat โ‰ˆ 3,500 kcal. A 500 kcal/day deficit ร— 7 days = 3,500 kcal/week.

The recommended amount of dietary protein needed to support muscle repair, immune function, and overall health.

Daily Protein (g) = Body Weight (kg) ร— Protein Factor
SymbolVariableUnit
Body WeightCurrent weightkg
Protein FactorActivity-dependent multiplierg/kg (0.8โ€“2.2)

Worked Example

Weight: 75 kgActivity: Strength training (1.8 g/kg)

Result: 135 g protein/day

75 ร— 1.8 = 135 g. Spread across 4 meals = ~34 g per meal.

Monthly Amortization Payment

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The process of spreading loan payments over time so that each payment covers both interest and principal, gradually reducing the balance to zero.

M = P ร— [r(1+r)^n] / [(1+r)^n โˆ’ 1]
SymbolVariableUnit
PPrincipal (loan amount)$
rMonthly interest ratedecimal (annual rate รท 12)
nTotal number of paymentsmonths

Worked Example

Loan: $250,000Rate: 6.5%/yr (0.5417%/mo)Term: 30 years (360 months)

Result: $1,580.17/month

M = 250,000 ร— [0.005417(1.005417)^360] / [(1.005417)^360 โˆ’ 1] โ‰ˆ $1,580.17.

Return on Investment

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Return on Investment โ€” a performance metric that measures the gain or loss from an investment relative to its cost, expressed as a percentage.

ROI = [(Current Value โˆ’ Cost) / Cost] ร— 100
SymbolVariableUnit
Current ValueCurrent value of investment$
CostOriginal investment cost$

Worked Example

Investment: $10,000Current Value: $13,500

Result: 35% ROI

ROI = [(13,500 โˆ’ 10,000) / 10,000] ร— 100 = 35%.

Future Value of Savings

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The projected growth of savings over time with regular contributions and compound interest.

FV = P(1+r)^n + PMT ร— [((1+r)^n โˆ’ 1) / r]
SymbolVariableUnit
PInitial deposit$
PMTRegular contribution$/period
rInterest rate per perioddecimal
nNumber of periodsmonths or years

Worked Example

Initial: $5,000Monthly: $500Rate: 5%/yrTerm: 10 years

Result: $85,624

Initial grows to $8,144; contributions add $77,480 with interest.

The projected increase in the value of invested capital over time through compound returns.

FV = PV ร— (1 + r/n)^(nร—t)
SymbolVariableUnit
PVPresent value (initial investment)$
rAnnual return ratedecimal
nCompounding frequencytimes per year
tTimeyears

Worked Example

Investment: $25,000Return: 8%/yrCompounding: MonthlyTime: 20 years

Result: $123,794

FV = 25,000 ร— (1 + 0.08/12)^(12ร—20) = 25,000 ร— 4.952 = $123,794.

Monthly Auto Loan Payment

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A secured loan used to purchase a vehicle, where the car serves as collateral.

M = P ร— [r(1+r)^n] / [(1+r)^n โˆ’ 1]
SymbolVariableUnit
PLoan amount (price โˆ’ down payment โˆ’ trade-in)$
rMonthly interest ratedecimal
nLoan termmonths (36, 48, 60, 72)

Worked Example

Vehicle Price: $35,000Down Payment: $5,000Rate: 5.9%Term: 60 months

Result: $580/month

Loan = $30,000. r = 0.059/12 = 0.00492. M = $30,000 ร— 0.00492(1.00492)^60 / [(1.00492)^60โˆ’1] โ‰ˆ $580.

CD Maturity Value

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A time-deposit savings product that pays a fixed interest rate for a fixed term, typically offering higher rates than regular savings accounts.

A = P ร— (1 + r/n)^(nร—t)
SymbolVariableUnit
PDeposit amount$
rAnnual interest ratedecimal
nCompounding frequencytimes per year
tTermyears

Worked Example

Deposit: $10,000APY: 5.0%Term: 12 monthsCompounding: Daily

Result: $10,513 at maturity

A = 10,000 ร— (1 + 0.05/365)^365 = 10,000 ร— 1.05127 = $10,513.

Equated Monthly Installment โ€” a fixed monthly payment that includes both principal and interest, commonly used for home, car, and personal loans.

EMI = [P ร— r ร— (1+r)^n] / [(1+r)^n โˆ’ 1]
SymbolVariableUnit
PPrincipal loan amountโ‚น or $
rMonthly interest rateannual rate รท 12
nTotal number of monthly paymentsmonths

Worked Example

Loan: โ‚น50,00,000Rate: 8.5%/yrTerm: 20 years (240 months)

Result: โ‚น43,391/month

r = 8.5/1200 = 0.00708. EMI = 50,00,000 ร— 0.00708 ร— (1.00708)^240 / [(1.00708)^240 โˆ’ 1] โ‰ˆ โ‚น43,391.

SIP Future Value

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Systematic Investment Plan โ€” a disciplined approach to investing a fixed amount regularly in mutual funds.

FV = PMT ร— [((1+r)^n โˆ’ 1) / r] ร— (1+r)
SymbolVariableUnit
PMTMonthly SIP amountโ‚น
rExpected monthly return ratedecimal (annual รท 12)
nTotal months of investmentmonths

Worked Example

Monthly SIP: โ‚น10,000Expected Return: 12%/yrDuration: 15 years

Result: โ‚น50.46 lakh

r = 0.01, n = 180. FV = 10,000 ร— [((1.01)^180 โˆ’ 1) / 0.01] ร— 1.01 โ‰ˆ โ‚น50,45,760.

Goods and Services Tax โ€” India's unified indirect tax on the supply of goods and services.

GST Amount = Base Price ร— (GST Rate / 100)
SymbolVariableUnit
Base PricePrice before taxโ‚น
GST RateApplicable GST slab% (5, 12, 18, or 28)

Worked Example

Base Price: โ‚น10,000GST Rate: 18%

Result: GST = โ‚น1,800 | Total = โ‚น11,800

โ‚น10,000 ร— 0.18 = โ‚น1,800 GST. Intra-state: โ‚น900 CGST + โ‚น900 SGST.

Progressive Tax Calculation

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A progressive tax levied by the government on individual or business earnings, with rates increasing as income rises through defined tax brackets.

Tax = ฮฃ (Income in bracket ร— Bracket rate)
SymbolVariableUnit
Taxable IncomeGross income minus deductions$
Bracket RatesUS 2024: 10%, 12%, 22%, 24%, 32%, 35%, 37%%

Worked Example

Taxable Income: $85,000 (Single filer, 2024)

Result: $14,260 federal tax

10% on first $11,600 + 12% on $11,601โ€“$47,150 + 22% on $47,151โ€“$85,000.

A consumption tax imposed by the government on the sale of goods and services, added at the point of purchase.

Tax Amount = Price ร— (Tax Rate / 100)
SymbolVariableUnit
PricePre-tax price of item$
Tax RateState + local tax rate%

Worked Example

Price: $49.99Tax Rate: 8.25%

Result: $4.12 tax | $54.11 total

$49.99 ร— 0.0825 = $4.12.

Population Standard Deviation

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A statistical measure of the amount of variation or dispersion in a dataset, indicating how spread out values are from the mean.

ฯƒ = โˆš[ฮฃ(xแตข โˆ’ ฮผ)ยฒ / N]
SymbolVariableUnit
xแตขEach data valueany
ฮผPopulation meansame as data
NPopulation sizecount

Worked Example

Data: 4, 8, 6, 5, 3

Result: ฯƒ โ‰ˆ 1.67

Mean = 5.2. Deviationsยฒ: 1.44+7.84+0.64+0.04+4.84 = 14.8. ฯƒ = โˆš(14.8/5) = 1.72.

The inverse operation of exponentiation โ€” log_b(x) answers "what power of b gives x?"

If b^y = x, then log_b(x) = y
SymbolVariableUnit
bBasenumber (usually 10, e, or 2)
xArgument (must be > 0)number
yExponent / resultnumber

Worked Example

Expression: logโ‚โ‚€(1000)

Result: 3

10ยณ = 1000, therefore logโ‚โ‚€(1000) = 3.

A mathematical notation indicating how many times a base number is multiplied by itself.

a^n = a ร— a ร— a ร— ... (n times)
SymbolVariableUnit
aBase numberany number
nExponent / powerinteger or real number

Worked Example

Expression: 2^10

Result: 1,024

2 ร— 2 ร— 2 ร— 2 ร— 2 ร— 2 ร— 2 ร— 2 ร— 2 ร— 2 = 1,024.

Fraction Operations

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A numerical quantity that represents part of a whole, written as a/b where a is the numerator and b is the denominator.

a/b + c/d = (ad + bc) / bd | a/b ร— c/d = ac / bd
SymbolVariableUnit
a/bFirst fractionratio
c/dSecond fractionratio

Worked Example

Problem: 3/4 + 2/5

Result: 23/20 = 1 3/20

(3ร—5 + 2ร—4) / (4ร—5) = (15 + 8) / 20 = 23/20.

Basic Probability

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The mathematical measure of the likelihood that an event will occur, expressed as a number between 0 and 1.

P(Event) = Favorable outcomes / Total outcomes
SymbolVariableUnit
P(Event)Probability of event0 to 1 (or 0% to 100%)
FavorableNumber of ways event can happencount
TotalTotal possible outcomescount

Worked Example

Event: Drawing a heart from a standard deck

Result: 13/52 = 0.25 = 25%

13 hearts out of 52 total cards. P = 13/52 = 1/4.

Quadratic Formula

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A second-degree polynomial equation of the form axยฒ + bx + c = 0.

x = (โˆ’b ยฑ โˆš(bยฒ โˆ’ 4ac)) / 2a
SymbolVariableUnit
aCoefficient of xยฒnumber (a โ‰  0)
bCoefficient of xnumber
cConstant termnumber

Worked Example

Equation: xยฒ โˆ’ 5x + 6 = 0 (a=1, b=โˆ’5, c=6)

Result: x = 3 or x = 2

x = (5 ยฑ โˆš(25โˆ’24)) / 2 = (5 ยฑ 1) / 2 โ†’ x = 3 or x = 2.

Binary to Decimal Conversion

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The base-2 numeral system using only digits 0 and 1, the foundation of all digital computing.

Decimal = ฮฃ(dแตข ร— 2^i)
SymbolVariableUnit
dแตขBinary digit at position i0 or 1
iPosition from right (starting at 0)integer

Worked Example

Binary: 11010

Result: 26 (decimal)

1ร—16 + 1ร—8 + 0ร—4 + 1ร—2 + 0ร—1 = 16 + 8 + 2 = 26.

Common Volume Formulas

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The amount of three-dimensional space enclosed within a closed surface, measured in cubic units.

Sphere: (4/3)ฯ€rยณ | Cylinder: ฯ€rยฒh | Cone: (1/3)ฯ€rยฒh | Box: lร—wร—h
SymbolVariableUnit
rRadiuslength units
hHeightlength units
l, wLength, Widthlength units

Worked Example

Shape: CylinderRadius: 5 cmHeight: 10 cm

Result: 785.4 cmยณ

V = ฯ€ ร— 5ยฒ ร— 10 = ฯ€ ร— 250 = 785.4 cmยณ.

Pythagorean Theorem

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A triangle with one 90-degree angle, where the relationship between sides follows the Pythagorean theorem.

aยฒ + bยฒ = cยฒ
SymbolVariableUnit
aOne leglength
bOther leglength
cHypotenuse (longest side, opposite 90ยฐ)length

Worked Example

Leg a: 3Leg b: 4

Result: Hypotenuse c = 5

c = โˆš(3ยฒ + 4ยฒ) = โˆš(9 + 16) = โˆš25 = 5. Classic 3-4-5 right triangle.

Grade Point Average โ€” a standardized measure of academic achievement calculated by averaging grade points earned across courses.

GPA = ฮฃ(Grade Points ร— Credit Hours) / ฮฃ(Credit Hours)
SymbolVariableUnit
Grade PointsNumeric value of letter gradeA=4, B=3, C=2, D=1, F=0
Credit HoursCourse credit weightcredits

Worked Example

Courses: A (3 cr) + B+ (4 cr) + A- (3 cr)

Result: 3.57 GPA

(4ร—3 + 3.3ร—4 + 3.7ร—3) / (3+4+3) = (12 + 13.2 + 11.1) / 10 = 3.63.

A fundamental law of electronics stating that voltage equals current times resistance (V = IR).

V = I ร— R
SymbolVariableUnit
VVoltagevolts (V)
ICurrentamperes (A)
RResistanceohms (ฮฉ)

Worked Example

Current: 2 AResistance: 10 ฮฉ

Result: Voltage = 20 V

V = 2 ร— 10 = 20 volts.

Simply Supported Beam (Center Point Load)

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The displacement of a structural beam under load, critical for ensuring safety and serviceability in construction.

ฮด_max = PLยณ / (48EI)
SymbolVariableUnit
PApplied loadN or lbs
LBeam spanm or ft
EModulus of elasticityPa or psi
IMoment of inertiamโด or inโด

Worked Example

Load: 10,000 NSpan: 6 mE: 200 GPa (steel)I: 8.33ร—10โปโถ mโด (W200ร—27)

Result: ฮด = 2.7 mm

ฮด = 10,000 ร— 6ยณ / (48 ร— 200ร—10โน ร— 8.33ร—10โปโถ) = 0.0027 m = 2.7 mm.

Mass per unit volume of a substance, expressed in kg/mยณ or g/cmยณ.

ฯ = m / V
SymbolVariableUnit
ฯDensitykg/mยณ
mMasskg
VVolumemยณ

Worked Example

Mass: 500 gVolume: 200 cmยณ

Result: 2.5 g/cmยณ

ฯ = 500 / 200 = 2.5 g/cmยณ. Denser than water (1.0 g/cmยณ) โ€” it sinks.

The energy possessed by an object due to its motion, proportional to mass and the square of velocity.

KE = ยฝmvยฒ
SymbolVariableUnit
mMasskg
vVelocitym/s

Worked Example

Mass: 1,500 kg (car)Velocity: 27 m/s (60 mph)

Result: 546,750 J โ‰ˆ 547 kJ

KE = ยฝ ร— 1,500 ร— 27ยฒ = 0.5 ร— 1,500 ร— 729 = 546,750 J.

A fundamental equation relating the pressure, volume, temperature, and amount of an ideal gas.

PV = nRT
SymbolVariableUnit
PPressureatm or Pa
VVolumeL or mยณ
nAmount of substancemoles
RGas constant0.0821 Lยทatm/(molยทK) or 8.314 J/(molยทK)
TTemperatureKelvin

Worked Example

n: 2 molT: 300 KP: 1 atm

Result: V = 49.3 L

V = nRT/P = 2 ร— 0.0821 ร— 300 / 1 = 49.26 L.

A rotational force that causes an object to rotate about an axis, measured in newton-meters (Nยทm) or foot-pounds (ftยทlb).

ฯ„ = F ร— d ร— sin(ฮธ)
SymbolVariableUnit
ฯ„TorqueNยทm or ftยทlb
FApplied forceN or lb
dDistance from pivot (moment arm)m or ft
ฮธAngle between force and lever armdegrees

Worked Example

Force: 50 NDistance: 0.3 mAngle: 90ยฐ

Result: 15 Nยทm

ฯ„ = 50 ร— 0.3 ร— sin(90ยฐ) = 50 ร— 0.3 ร— 1 = 15 Nยทm.

Exponential Decay

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The time required for a quantity to reduce to half its initial value, commonly used in nuclear physics and pharmacology.

N(t) = Nโ‚€ ร— (ยฝ)^(t/tยฝ)
SymbolVariableUnit
Nโ‚€Initial quantityany
N(t)Remaining quantity at time tsame as Nโ‚€
tElapsed timesame as tยฝ
tยฝHalf-lifeyears, hours, etc.

Worked Example

Initial: 100 gHalf-life: 5,730 years (Carbon-14)Time: 11,460 years

Result: 25 g remaining

100 ร— (ยฝ)^(11,460/5,730) = 100 ร— (ยฝ)ยฒ = 100 ร— 0.25 = 25 g.

The analysis of fluid movement through pipes, considering flow rate, velocity, pressure drop, and friction losses.

Q = A ร— v = ฯ€(d/2)ยฒ ร— v
SymbolVariableUnit
QVolumetric flow ratemยณ/s or GPM
ACross-sectional area of pipemยฒ
vAverage flow velocitym/s
dPipe internal diameterm

Worked Example

Diameter: 4 inches (0.1016 m)Velocity: 2 m/s

Result: 0.0162 mยณ/s โ‰ˆ 257 GPM

A = ฯ€(0.0508)ยฒ = 0.00811 mยฒ. Q = 0.00811 ร— 2 = 0.01621 mยณ/s.

Cubic Meter โ€” the standard unit of measurement for shipping volume in international freight and logistics.

CBM = Length(m) ร— Width(m) ร— Height(m)
SymbolVariableUnit
LengthPackage lengthmeters
WidthPackage widthmeters
HeightPackage heightmeters

Worked Example

Length: 1.2 mWidth: 0.8 mHeight: 1.0 m

Result: 0.96 CBM

1.2 ร— 0.8 ร— 1.0 = 0.96 mยณ. For 10 such boxes: 9.6 CBM total.

Chargeable Weight

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The total charge for shipping goods from origin to destination, based on weight, volume, distance, and mode of transport.

Chargeable Weight = max(Actual Weight, Volumetric Weight)
SymbolVariableUnit
Actual WeightPhysical weight of shipmentkg
Volumetric WeightCBM ร— factor (e.g., 167 for air)kg
RateCarrier rate per kg or per CBM$/kg or $/CBM

Worked Example

Actual: 500 kgVolume: 4 CBMAir Factor: 167Rate: $2.50/kg

Result: $1,670 (charged on volumetric: 668 kg)

Volumetric = 4 ร— 167 = 668 kg > 500 kg actual. Cost = 668 ร— $2.50 = $1,670.

Area Calculation

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The area of a space measured in square feet, commonly used in US real estate and construction to describe floor area.

Area = Length ร— Width (rectangular rooms)
SymbolVariableUnit
LengthRoom lengthfeet
WidthRoom widthfeet

Worked Example

Length: 15 ftWidth: 12 ft

Result: 180 sq ft

15 ร— 12 = 180 square feet. For irregularly shaped rooms, divide into rectangles and sum.

The calculation of materials needed for a roof based on area, pitch, and material type.

Roof Area = Footprint Area ร— Pitch Multiplier
SymbolVariableUnit
FootprintBuilding length ร— widthsq ft
Pitch Multiplier4/12โ†’1.054, 6/12โ†’1.118, 8/12โ†’1.202, 12/12โ†’1.414factor

Worked Example

Footprint: 2,000 sq ftPitch: 6/12

Result: 2,236 sq ft โ†’ 22.4 roofing squares

2,000 ร— 1.118 = 2,236 sq ft รท 100 = 22.36 squares.

The calculation of posts, rails, and pickets needed for a fence based on linear footage and fence type.

Posts = (Total Length / Post Spacing) + 1
SymbolVariableUnit
Total LengthFence perimeterlinear feet
Post SpacingDistance between postsfeet (typically 6 or 8)

Worked Example

Length: 150 linear feetPost Spacing: 8 ftType: 6ft privacy

Result: 20 posts, 38 rails, 300 pickets

Posts: (150/8)+1 = 19.75 โ†’ 20. Rails: 2 per section ร— 19 = 38. Pickets: 150/0.5 = 300 (at 6" spacing).

The calculation of decking boards, posts, and hardware needed for an outdoor deck construction project.

Boards = Deck Area / Board Coverage Area ร— Waste Factor
SymbolVariableUnit
Deck AreaTotal deck surfacesq ft
Board CoverageBoard width ร— length (typical: 5.5" ร— 12ft = 5.5 sq ft)sq ft per board
Waste FactorExtra for cuts and waste1.10โ€“1.15 (10โ€“15%)

Worked Example

Deck Size: 12ร—16 ft = 192 sq ftBoard: 5.5" ร— 12ft

Result: 39 boards (with 10% waste)

192 / 5.5 = 34.9 boards ร— 1.10 waste = 38.4 โ†’ 39 boards.

Flooring Material

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The calculation of flooring materials needed based on room area, accounting for waste from cuts and layout patterns.

Material = Room Area ร— Waste Factor
SymbolVariableUnit
Room AreaTotal floor areasq ft
Waste FactorExtra for cuts, damage, and future repairs1.10 (10%) standard, 1.15 for diagonal

Worked Example

Room: 350 sq ftPattern: Standard (10% waste)

Result: Order 385 sq ft of flooring

350 ร— 1.10 = 385 sq ft. For 20-sq-ft boxes: 385/20 = 19.25 โ†’ 20 boxes.

Electricity Cost

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The cost to operate an electrical device, based on its wattage, usage time, and the local electricity rate.

Cost = (Watts ร— Hours / 1000) ร— Rate per kWh
SymbolVariableUnit
WattsDevice power consumptionW
HoursDaily usage timehours
RateElectricity price$/kWh

Worked Example

Device: 1,500W space heaterUsage: 8 hours/dayRate: $0.13/kWh

Result: $1.56/day โ†’ $46.80/month

(1,500 ร— 8 / 1,000) ร— 0.13 = 12 kWh ร— $0.13 = $1.56/day.

The total fuel expense for a trip based on distance, vehicle fuel efficiency, and fuel price.

Cost = (Distance / MPG) ร— Price per Gallon
SymbolVariableUnit
DistanceTrip distancemiles
MPGVehicle fuel efficiencymiles per gallon
PriceFuel cost$/gallon

Worked Example

Trip: 500 milesMPG: 28Gas Price: $3.50/gal

Result: $62.50

500/28 = 17.86 gallons ร— $3.50 = $62.50.

The distance a vehicle can travel per unit of fuel, measured in miles per gallon (MPG) or kilometers per liter.

MPG = Miles Driven / Gallons Used
SymbolVariableUnit
MilesDistance driven since last fill-upmiles
GallonsFuel consumedgallons

Worked Example

Miles: 320 milesGallons: 11.5 gallons

Result: 27.8 MPG

320 / 11.5 = 27.83 MPG.

A tool to calculate the gratuity amount and total bill, optionally splitting among multiple diners.

Tip = Bill ร— (Tip% / 100)
SymbolVariableUnit
BillPre-tax bill amount$
Tip%Tip percentage% (typically 15โ€“25%)

Worked Example

Bill: $85.00Tip%: 20%Split: 4 people

Result: Tip: $17.00 | Total: $102.00 | Per person: $25.50

$85 ร— 0.20 = $17. Total = $102. Each of 4 pays $25.50.

Concrete Project Cost

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The estimated cost for a concrete project based on volume needed, concrete type, and delivery/labor costs.

Cost = Volume (cubic yards) ร— Price per cubic yard
SymbolVariableUnit
VolumeTotal concrete neededcubic yards
PriceReady-mix cost per yard$/cubic yard

Worked Example

Slab: 20ร—20 ft, 4" thickPrice: $150/ydยณ

Result: ~$740 for concrete only

20ร—20ร—0.333 = 133.3 cu ft รท 27 = 4.94 ydยณ ร— $150 = $741.

Data Size Conversion

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A measure of digital information expressed in bits, bytes, and their multiples (KB, MB, GB, TB).

Each step = ร—1,024 (binary) or ร—1,000 (decimal/SI)
SymbolVariableUnit
Byte8 bitsB
KB1,024 bytesKilobyte
MB1,024 KBMegabyte
GB1,024 MBGigabyte
TB1,024 GBTerabyte

Worked Example

Size: 5 GB

Result: 5,120 MB = 5,242,880 KB = 5,368,709,120 Bytes

5 ร— 1,024 = 5,120 MB. 5,120 ร— 1,024 = 5,242,880 KB.